Rep:Mod:cn22093
The Cope Rearrangement
The first part of this investigation, an analysis of the well-known Cope rearrangement, is an opportunity to run through and practice the computational methods which will then be used in the later stages. The Cope rearrangement, named after chemist Arthur C. Cope, is an example of the [3,3]-sigmatropic shift pericyclic reaction.[1]
Optimising Reactants and Products
Set out in the table below are a series of possible conformations of 1,5-hexadiene, the molecule which undergoes the Cope rearrangement. The conformations differ in the values of their dihedral angles. The central four carbons have been manipulated in such a way so as to give dihedral angles of either 60o (gauche) or 180o (antiperiplanar). By observation of the Fischer projections below it would be possible to make a qualitative assumption that the gauche conformation may be more sterically strained and therefore less stable.
The dihedral angles of the four terminal carbons have also been systematically varied and the models optimised using the fairly low level but computationally inexpensive Hartree-Fock/3-21G combination. The data includes the optimised energy, the energies relative to the most stable conformer (gauche 3) and the point group symmetries.
| Conformer | Structure (Jmol) | Energy (Hartree) | Relative Energy (kcal/mol) | Point Group |
|---|---|---|---|---|
| Anti 1 |  |
-231.69260 | 0.04 | C2 |
| Anti 2 |  |
-231.69254 | 0.08 | Ci |
| Anti 3 |  |
-231.68907 | 2.25 | C2h |
| Anti 4 |  |
-231.69097 | 1.06 | C1 |
| Gauche 1 |  |
-231.68772 | 3.10 | C2 |
| Gauche 2 |  |
-231.69167 | 0.62 | C2 |
| Gauche 3 |  |
-231.69266 | 0.00 | C1 |
| Gauche 4 |  |
-231.69153 | 0.71 | C2 |
| Gauche 5 |  |
-231.68962 | 1.91 | C1 |
| Gauche 6 |  |
-231.68916 | 2.20 | C1 |
The above results have revealed gauche 3 to be the most stable conformer with anti 1 and anti 2 very close behind. As these three have been predicted to be the most stable using the HF/3-21G level it was deemed useful to reoptimise these three at a higher level of theory to see if there would be any variation in their geometries or energies. The chosen method and basis set was B3LYP/6-31G(d), the results for this investigation are provided below.
| Conformer | Structure (Jmol) | Energy (Hartree) | Relative Energy (kcal/mol) | Point Group | |||
|---|---|---|---|---|---|---|---|
| Anti 1 |
|
-234.61180 | 0.00 | C2 | |||
| Anti 2 |
|
-234.61170 | 0.06 | Ci | |||
| Gauche 3 |
|
-234.61133 | 0.30 | C1 |
Two clear observations can be made from these results. All electronic energies have decreased by around 3 Hartree, or 2000kcal/mol. The change in energies has also changed their relative stabilities, anti 1 has now been predicted as being the most stable. Optimising at this level has not markedly changed the geometry of the model, the bond lengths have decreased slightly and the dihedral angles have increased slightly.
Having reoptimised the anti 2 conformer it would also be helpful to carry out a frequency analysis at the same B3LYP/6-31G* level to establish whether a minimum has been obtained, indicated by the lack of any large negative frequencies. In this case none were observed and it can be concluded that at this level a fully optimised geometry has been obtained. The IR spectrum of this frequency analysis has been presented below.
Although the B3LYP/6-31G* optimisation has revealed anti 1 to be the more stable conformer, some key thermochemistry values for the anti 2 conformer have been found in order to practice finding these values for later investigations. These have been obtained from the text based log file after frequency analyses at 0K and 298.15K.
298.15K
Zero-point correction= 0.142491 (Hartree/Particle) Thermal correction to Energy= 0.149847 Thermal correction to Enthalpy= 0.150791 Thermal correction to Gibbs Free Energy= 0.110882 Sum of electronic and zero-point Energies= -234.469212 Sum of electronic and thermal Energies= -234.461856 Sum of electronic and thermal Enthalpies= -234.460912 Sum of electronic and thermal Free Energies= -234.500821
0K
Zero-point correction= 0.142928 (Hartree/Particle) Thermal correction to Energy= 0.142928 Thermal correction to Enthalpy= 0.142928 Thermal correction to Gibbs Free Energy= 0.142928 Sum of electronic and zero-point Energies= -234.468775 Sum of electronic and thermal Energies= -234.468775 Sum of electronic and thermal Enthalpies= -234.468775 Sum of electronic and thermal Free Energies= -234.468775
In order to carry out calculations at a later stage, it is important to explain some of these values. The sum of electronic and zero-point energies is the potential energy at 0 K including the zero-point vibrational energy, this value will be taken for calculating activation energies computed at 0K. The sum of electronic and thermal energies is the potential energy at 298.15 K and 1 atm of pressure, this includes translational, rotational, and vibrational energy contributions. This value will be used when calculating activation energies at 298.15K.
Optimising Chair and Boat Transition Structures
The Cope rearrangement is known to go through two possible transition state structures, these are the chair and boat forms, similar to the two possible cyclohexane conformations with which they share names. In cyclohexane the chair is the most stable wherease the boat form is destabilised due to torsional strain and steric clashing between flagpole hydrogens. This might indicate the relationship between the two transition states of the Cope rearrangement, computational techniques have been used to further explore this relationship.
Both transition states consist of two allyl (CH2CHCH2) fragments positioned in particular orientation to one another as set out below.
CHAIR
Two different methods were used to optimise the chair transition structure, both were at the HF/3-21G level of theory. The first method was a simple TS(Berny) optimisation, this relies on the allyl fragments already being positioned appropriately prior to optimisation. The second method is the frozen coordinate method where the C-C bonds that will form in the product are frozen, while the rest of the model is optimised to a minimum. Then after this first optimisation the coordinates can be unfrozen and a TS(Berny) optimisation carried out.
After the first optimisation a frequency analysis was carried out to confirm presence of an imaginary frequency (given by a negative value) corresponding to the [3,3]-sigmatropic shift. This was found, a value of -817.92cm-1 was recorded and the animation confirms that this vibration is that of the Cope rearrangement. The σ C-C bonds to be formed have come out as 2.02049 and 2.0247Å which, accounting for error of the calculations, shows good equivalence as expected.
The frozen coordinate method gave a very similar structure to that of the first optimisation except with C-C bond lengths of 2.01985 and 2.01988Å. Between the two methods this is a discrepancy of approximately 0.0006Å which is negligible once again when considering the error limits of the calculation method.
Shown below are the graphs obtained from two IRC studies. An IRC or intrinsic reaction coordinate is a computational technique used to follow the reaction path from a maximum along the potential energy surface (transition state) to its local minimum (reactant/product, in this case they are equivalent). This is with the aim of establishing which conformers modelled earlier on in the report are connected by these transition states.
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As can be seen from the results above, the initial IRC method (graphs on the left) where the force constants have only been calculated once has not led to an adequate minimum. The RMS gradient value is 0.00117605 which is too large to be considered that of a global minimum. The IRC calculation was then run again, this time calculating force constants at every step. This resulted in the graphs above on the right which have given a more adequate minimum, and an RMS gradient value of 0.00001617. This IRC also produced a series of geometry steps, with the final structure indicating the reactant/product conformer. From inspection, the conformer most closely resembles gauche 2.
BOAT
The boat transition state was optimised using a different approach, the QST2 method. Instead of starting from an approximation of the transition state structure, this method starts with the reactants and products and interpolates between them to try and find the transition state.
The first attempt started with the optimised anti 2 conformer from the first section. For the QST2 method to work properly the reactants and products must be accurately modelled and the numbering of all the atoms of both structures must properly reflect their relationship. For example, a terminal carbon in the product will end up as an inside carbon, this specific information has therefore been computed before the calculation was run. The particular numbering used for this run has been shown below.
As can be seen from the structure obtained from the optimisation, the method has not worked properly and has given a structure more resembling the chair transition structure. The method has not accounted for the possibility of the allyl fragments rotating about the axis between the central carbons of both fragments.
In the next attempt the reactant and product structures were manipulated in such a way to more closely resemble the desired boat transition state. These manipulations reduced the dihedral angle of the central four carbons to 0o and the angles of the inside carbons 2,3,4 and 5,4,3 to 100o. This gave the reactants and products below, still numbered appropriately.
Optimising this time has given the correct boat transition state structure. Once again carrying out a frequency analysis reveals just one imaginary frequency at -840.63cm-1. Animating the vibrational mode shows it to correspond to the Cope rearrangement as expected.
From the results of the IRC study of the chair transition structure, the same 'calculate force constants always' method has been applied to the boat. This has produced the results below.
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An adequate minimum has been obtained and the final molecule of the geometry step sequence closely resembles the gauche 2 aswell.
Activation Energies
As was done for the anti 2 conformer, thermochemistry values have been obtained from frequency analyses at both 0K and 298.15K. To make for even more elucidating comparisons, this analysis has been carried out for the two levels of theory HF/3-21G and DFT B3LYP/6-31G*. As the gauche 3 was found to be the most stable conformer for the HF optimisations, this has been chosen as the conformer for calculating activation energies at this level. Anti 1 is the analogous choice for calculating activation energies at the DFT level of theory.
HF/3-21G (in Hartrees)
| Electronic Energy | Sum of electronic and zero-point energies (at 0K) |
Sum of electronic and thermal energies (at 298.15K) | |
|---|---|---|---|
| Chair | -231.619322 | -231.464793 | -231.460163 |
| Boat | -231.602800 | -231.450451 | -231.445291 |
| Gauche 3 | -231.692661 | -231.539015 | -231.532647 |
DFT B3LYP/6-31G(d) (in Hartrees)
| Electronic Energy | Sum of electronic and zero-point energies (at 0K) |
Sum of electronic and thermal energies (at 298.15K) | |
|---|---|---|---|
| Chair | -234.556983 | -234.414488 | -234.409010 |
| Boat | -234.543093 | -234.401901 | -234.396005 |
| Anti 1 | -234.611800 | -234.468848 | -234.461965 |
Summary of activation energies (in kcal/mol)
| HF/3-21G (at 0K) |
HF/3-21G (at 298.15K) |
DFT B3LYP/6-31G* (at 0K) | DFT B3LYP/6-31G* (at 298.15K) | Experimental (at 0K) | |
|---|---|---|---|---|---|
| ΔE - Chair | 46.58 | 45.48 | 34.11 | 33.23 | 33.5 ± 0.5 |
| ΔE - Boat | 55.57 | 54.82 | 42.01 | 41.39 | 44.7 ± 2.0 |
Both calculation methods show the chair transition state to be more stable than the boat, deduced dues to it having a smaller activation energy. This is in agreement with the experimental data and confirms as predicted that the chair is the true transition state of the Cope rearrangement. The trend is also unchanged by temperature. As previously discussed, the chair will be more stable due to less torsional strain and less steric clashing between hydrogens. There are however secondary orbital effects that are likely to play a part in destabilising the boat transition state. In terms of computational methods, it is evident that the DFT method has produced activation energies that are more in line with the literature which is some evidence confirming that it is a higher level and more accurate method.
The Diels Alder Cycloaddition
The Diels Alder reaction is another example of a pericyclic reaction[2], specifically of the cycloaddition sub-type. The reaction is between a diene and a dienophile to produce a cyclohexene ring. In this report, investigation into the Diels Alder reaction will start with the simplest example (cis-butadiene + ethene → cyclohexene) and then move onto one of the more common examples (cyclohexa-1,3-diene + maleic anhydride). Computational techniques established in the previous section will be used for the investigation.
Prototypical Reaction: Cis-butadiene & Ethene
Models of the two reactants cis-butadiene and ethene were optimised at the semi-empirical AM1 level of theory, 3D graphical representations of the HOMO and LUMO have been generated for both structures and displayed below. The symmetry of these MOs can be assessed with respect to the two planes shown in the diagram to the right. The HOMO of cis-butadiene can be seen to be antisymmetric with respect to both horizontal and vertical planes. The LUMO is then shown to be antisymmetric with respect to the horizontal plane and symmetric with respect to the vertical. Similarly for the ethene MOs, the HOMO is symmetric with respect to both and the LUMO antisymmetric with respect to both.
| Cis-butadiene | Ethene | |
|---|---|---|
| LUMO |  |
|
| HOMO |  |
|
Seeing as the frozen coordinate method was successful in optimising transition state structures in the Cope rearrangement it will be attempted to optimise the transition state structure in the prototypical Diels Alder reaction using this method again. The lengths between the ring-forming carbons was set to 2.2Å in the redundant coordinate editor and the model was then optimised to a minimum using the AM1 level. Once this completed successfully the bonds were unfrozen using the Derivative option in the editor and a TS(Berny) optimisation carried out. This produced the following
structure. The bond lengths were measured, the partly formed σ C-C bond lengths were both 2.12Å, all double bonds of both molecules were measured at 1.38Å and the single bond in butadiene 1.40Å. Typical sp3-sp3 and sp2-sp2 C-C bond lengths are 1.54 and 1.33Å respectively[3]. sp2 double bonds are generally shorter than sp3 single bonds due to having greater s-character. The bond lengths of the optimised structure show that the double bonds are longer than typical and the single bond is shorter than typical. This means the transition state has accounted for electron pushing, the single bond is gaining electrons which will increase the s-character shortening and vice versa for the double bonds. The partly formed C-C bond is much greater in length than the typical sp3 bond length because there is not very much electron density between the carbons. The distance is however less than the sum of two C vdW radii (1.7Å)[4]. This means there will be some attractive interaction between the vdW radii of the terminal carbons.
After optimising, a frequency analysis was carried out at the semi-empirical AM1 level to examine the vibrational frequencies. An imaginary frequency was observed at -955.88cm-1 while the lowest positive frequency was observed at 147.02cm-1. As can be seen from the animation, the negative frequency corresponds to the formation of cyclohexene with bonds being made and broken in a synchronous fashion. In other words the vibrational mode shows the reaction to be concerted, predicted for a pericyclic reaction. The lowest positive vibrational mode is asynchronous and can be described as a cyclisation.
| Cycloaddition transition state | |
|---|---|
| LUMO |  
|
| HOMO |  
|
Qualitatively speaking, the HOMO and LUMO of the transition state show overlapping of MOs from the original reactants. This can be related back to the symmetry analysis of the reactant MOs. The HOMO of the transition state clearly shows overlap of the HOMO of the cis-butadiene and the LUMO of ethene. This interaction is allowed because they are both antisymmetric with respect to both horizontal and vertical planes, as is the resulting MO of the transition state. The LUMO of the transition state then shows overlap of the LUMO of cis-butadiene and the HOMO of ethene. This interaction is allowed because they are both symmetric with respect to the vertical plane, again the same symmetry property is maintained in the resulting MO.
Substituted Reagents: Cyclohexa-1,3-diene & Maleic Anhydride
Endo
|
Exo
|
|---|
The following reaction is another example of a Diels Alder cycloaddition reaction, this time the reactants are substituted though which is likely to effect the overall reactivity. Unlike the first reaction this reaction is regioselective, having two possible regioisomers, endo and exo shown in the reaction scheme below. The two possible transition states were modelled and optimised using the AM1 level of theory again and the frozen coordinate method. This produced the transition states shown to the left with energies of -0.0515 Hartree for the endo and -0.0504 for the exo. This predicts the endo transition state to be more stable and therefore if the reaction is under kinetic control then the endo isomer will predominate. Looking more closely at the geometries, it would be likely that the exo would be slightly more strained due to steric repulsions between the hydrogens and the anhydride. The results so far support this idea.
Frequency analysis has given one endo negative frequency and one exo negative frequency both corresponding to the expected synchronous and therefore concerted cycloadditions. The
endo lowest positive frequency and exo lowest positive frequency are also the expected asynchronous stretches which don't refer to formation of the products.
| Endo | Exo | |
|---|---|---|
| LUMO+1 |   |
 
|
| LUMO |   |
 
|
| HOMO |   |
 
|
Some of the most relevant MOs of both transition states have been visualised to try and come to any conclusions regarding orbital overlap. The HOMOs of both transition states show bonding primary orbital overlap between the p orbitals of the double bond on the anhydride and the p orbitals of the . A secondary orbital overlap would be expected for the endo transition state as shown in the diagram below. However, presumably because AM1 is too low level, there seems to little to no bonding interactions between the carbonyl carbons and the ring. This secondary orbital overlap serves to stabilise the endo transition state, again confirming the endo product to be the kinetic product. If the endo product is the kinetic product it is likely that the exo is the thermodynamic product. An IRC has therefore been carried out to support this prediction.
| Endo | Exo | |
|---|---|---|
| Intrinsic Reaction Coordinate |  |
|
| Animation of geometry steps |  |
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Just by inspection of the top IRC graph it can be seen that the exo product is the more stable, having an energy of -0.16017026 Hartree as opposed to -0.15990908 Hartree. This predicts the exo product to be more stable by approximately 0.16 kcal/mol, a relatively significant amount. Despite this the endo product has been known to form almost exclusively and this can be attributed to the stability of its transition state. These stabilisations are brought about by destabilising repulsions of the exo as well as stabilising secondary orbital overlaps present in the endo but absent in the exo.
Conclusion
A number of possible conformers of 1,5-hexadiene were optimised at HF/3-21G level and gauche 3 was found to be the most stable. Optimisation at a higher level DFT B3LYP/6-31G* then revealed the anti 1 to be the most stable. Chair and boat transition structures of the Cope rearrangement were optimised using the same HF/3-21G level but with different methods - these were the frozen coordinate method and the QST2 method. Both methods appeared to work with relative success. Intrinsic reaction coordinate calculation helped possibly identify the conformer which the transition states connected. One of the more interesting applications of computational techniques was in calculating activation energies of the two transition states, this was done at two different temperatures as well as using two different basis sets. The results confirmed the chair to be the more stable transition state and the DFT method to be more in line with experimental data.
The second part of the investigation was into two separate Diels Alder reactions, the simple formation of cyclohexene and the regioselective reaction between substituted reactants . Optimisations for this section were carried out using semi-empirical AM1 level of theory and the frozen coordinate method. This again worked well, however a higher level of theory would have maybe more explicitly revealed the secondary orbital effects described. Visualisation of the MOs allowed discussion into allowed and forbidden interactions, based on sharing symmetry properties. Frequency analysis also showed the vibrational modes corresponding to formation of products. The computational techniques used also help to establish that the endo product is the kinetic product of the reaction and the exo product is the thermodynamic product.
References
- ↑ Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.
- ↑ H. Rzepa, Pericylic reactions http://www.ch.ic.ac.uk/local/organic/pericyclic
- ↑ Fox, MA and JK Whitesell. Organische Chemie. 1994. Spektrum
- ↑ Stoicheff, B.P., Tetrahedron, 1962, 17, 135-145
- ↑ Ian Fleming, Frontier orbitals and organic chemical reactions, John Wiley & Sons, 2006, 106-107
Dspace Links
| Cope (reactants and products | Cope (chair and boat) | Diels Alder (prototypic) | Diels Alder (substituted) |
|---|---|---|---|
| Anti 1 HFDOI:10042/to-11535 Anti 2 HFDOI:10042/to-11536 Gauche 3 HFDOI:10042/to-11541 Gauche 4 HF DOI:10042/to-11559 Gauche 5 HF DOI:10042/to-11605 |
Chair (frozen coord) DOI:10042/to-11574 Chair IRC DOI:10042/to-11604 Boat HF (successful) DOI:10042/to-11567 Boat DFT, 298K DOI:10042/to-11602 Boat IRC DOI:10042/to-11606 Boat DFT, 0K DOI:10042/to-11601 |
Cis-but AM1 DOI:10042/to-11592 Ethene AM1 DOI:10042/to-11561 Cyclo TS (frozen coord) DOI:10042/to-11542 Cyclo freq DOI:10042/to-11563 |
Endo AM1 (frozen coord) DOI:10042/to-11579 Exo AM1 (frozen coord) DOI:10042/to-11600 Endo freq DOI:10042/to-11596 Exo freq DOI:10042/to-11597 Endo IRC DOI:10042/to-11598 Exo IRC DOI:10042/to-11599 |